The given equation is:

\[
\frac{1}{\cos^2(x)} + \frac{1}{\sin^2(x)} = 4
\]

This equation involves trigonometric functions. Let's break it down and solve it.

### Step 1: Simplify the expression
We can rewrite the terms in terms of secant and cosecant functions:
\[
\sec^2(x) + \csc^2(x) = 4
\]
where:
\[
\sec(x) = \frac{1}{\cos(x)} \quad \text{and} \quad \csc(x) = \frac{1}{\sin(x)}
\]

### Step 2: Use trigonometric identities
We know the identity:
\[
\sec^2(x) = 1 + \tan^2(x)
\quad \text{and} \quad 
\csc^2(x) = 1 + \cot^2(x)
\]
Substituting these into the equation, we get:
\[
1 + \tan^2(x) + 1 + \cot^2(x) = 4
\]
Simplifying:
\[
2 + \tan^2(x) + \cot^2(x) = 4
\]

### Step 3: Solve for \(\tan^2(x)\) and \(\cot^2(x)\)
Subtract 2 from both sides:
\[
\tan^2(x) + \cot^2(x) = 2
\]

Now, solve for \(\tan(x)\) and \(\cot(x)\). We know the relation between \(\tan(x)\) and \(\cot(x)\):
\[
\cot(x) = \frac{1}{\tan(x)}
\]
Substitute this in:
\[
\tan^2(x) + \frac{1}{\tan^2(x)} = 2
\]
Multiply through by \(\tan^2(x)\):
\[
\tan^4(x) + 1 = 2\tan^2(x)
\]

### Step 4: Set up a quadratic equation
Let \( y = \tan^2(x) \), so the equation becomes:
\[
y^2 - 2y + 1 = 0
\]
This can be factored as:
\[
(y - 1)^2 = 0
\]
Thus, \( y = 1 \).

### Step 5: Solve for \(x\)
Since \( y = \tan^2(x) \), we have:
\[
\tan^2(x) = 1
\]
Taking the square root:
\[
\tan(x) = \pm 1
\]
This implies that:
\[
x = \frac{\pi}{4} + n\pi \quad \text{for some integer } n
\]

### Final Solution:
The solutions for \(x\) are:
\[
x = \frac{\pi}{4} + n\pi, \quad \text{where } n \in \mathbb{Z}
\]

---

Do you want further details on any part of the solution? Here are 5 related questions for practice:

1. What is the relation between \(\tan(x)\) and \(\cot(x)\)?
2. Can you derive \(\sec^2(x) + \csc^2(x)\) from basic trigonometric identities?
3. How do you solve trigonometric quadratic equations like \(\tan^4(x) - 2\tan^2(x) + 1 = 0\)?
4. What are the general solutions for \(\tan(x) = 1\) and \(\tan(x) = -1\)?
5. How would the solution change if the right-hand side of the equation were different, e.g., 5 instead of 4?

**Tip:** Always check if trigonometric identities can simplify the problem before attempting to solve complex equations.

Explore how to solve the trigonometric equation cos^2(x) + sin^2(x) = 4 by using trigonometric identities and quadratic equations. This solution walks through steps involving secant, cosecant, and tangent functions, suitable for Grades 10-12 students.

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